mixedcos

Percentage Accurate: 66.7% → 97.9%

Time: 9.3s

Alternatives: 12

Speedup: 2.3×

• 32.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

Java

public static double code(double x, double c, double s) {
	return Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

Julia

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

Java

public static double code(double x, double c, double s) {
	return Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

Julia

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \cos \left(x\_m + x\_m\right)\\ t_1 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 9 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1}{t\_1}}{t\_1}\\ \mathbf{elif}\;x\_m \leq 10^{+167}:\\ \;\;\;\;\frac{t\_0}{\left(c\_m \cdot s\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(x\_m \cdot c\_m\right) \cdot \left(x\_m \cdot c\_m\right)\right) \cdot \left(s\_m \cdot s\_m\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (cos (+ x_m x_m))) (t_1 (* c_m (* x_m s_m))))
   (if (<= x_m 9e-86)
     (/ (/ 1.0 t_1) t_1)
     (if (<= x_m 1e+167)
       (/ t_0 (* (* c_m s_m) (* s_m (* x_m (* x_m c_m)))))
       (/ t_0 (* (* (* x_m c_m) (* x_m c_m)) (* s_m s_m)))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m + x_m));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 9e-86) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1e+167) {
		tmp = t_0 / ((c_m * s_m) * (s_m * (x_m * (x_m * c_m))));
	} else {
		tmp = t_0 / (((x_m * c_m) * (x_m * c_m)) * (s_m * s_m));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((x_m + x_m))
    t_1 = c_m * (x_m * s_m)
    if (x_m <= 9d-86) then
        tmp = (1.0d0 / t_1) / t_1
    else if (x_m <= 1d+167) then
        tmp = t_0 / ((c_m * s_m) * (s_m * (x_m * (x_m * c_m))))
    else
        tmp = t_0 / (((x_m * c_m) * (x_m * c_m)) * (s_m * s_m))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m + x_m));
	double t_1 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 9e-86) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x_m <= 1e+167) {
		tmp = t_0 / ((c_m * s_m) * (s_m * (x_m * (x_m * c_m))));
	} else {
		tmp = t_0 / (((x_m * c_m) * (x_m * c_m)) * (s_m * s_m));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m + x_m))
	t_1 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 9e-86:
		tmp = (1.0 / t_1) / t_1
	elif x_m <= 1e+167:
		tmp = t_0 / ((c_m * s_m) * (s_m * (x_m * (x_m * c_m))))
	else:
		tmp = t_0 / (((x_m * c_m) * (x_m * c_m)) * (s_m * s_m))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = cos(Float64(x_m + x_m))
	t_1 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 9e-86)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	elseif (x_m <= 1e+167)
		tmp = Float64(t_0 / Float64(Float64(c_m * s_m) * Float64(s_m * Float64(x_m * Float64(x_m * c_m)))));
	else
		tmp = Float64(t_0 / Float64(Float64(Float64(x_m * c_m) * Float64(x_m * c_m)) * Float64(s_m * s_m)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m + x_m));
	t_1 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 9e-86)
		tmp = (1.0 / t_1) / t_1;
	elseif (x_m <= 1e+167)
		tmp = t_0 / ((c_m * s_m) * (s_m * (x_m * (x_m * c_m))));
	else
		tmp = t_0 / (((x_m * c_m) * (x_m * c_m)) * (s_m * s_m));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 9e-86], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$95$m, 1e+167], N[(t$95$0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.9999999999999995e-86

   1. Initial program 64.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 67.6

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right) \cdot \left(x \cdot x\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      13. swap-sqr N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   7. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}} \]

   if 8.9999999999999995e-86 < x < 1e167

   1. Initial program 71.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 98.5

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 98.5%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      6. lift-/.f64 98.5

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      8. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      9. lift-+.f64 98.5

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      15. lower-*.f64 98.5

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      20. lift-*.f64 97.0

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

   6. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(x \cdot c\right) \cdot \left(s \cdot x\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(\left(x \cdot c\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(\left(x \cdot c\right) \cdot x\right) \cdot s\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(\left(x \cdot c\right) \cdot x\right) \cdot s\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(\left(\color{blue}{\left(c \cdot x\right)} \cdot x\right) \cdot s\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(\color{blue}{\left(\left(c \cdot x\right) \cdot x\right)} \cdot s\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(\left(\color{blue}{\left(x \cdot c\right)} \cdot x\right) \cdot s\right)} \]

      10. lower-*.f64 93.8

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(\left(\color{blue}{\left(x \cdot c\right)} \cdot x\right) \cdot s\right)} \]

   8. Applied rewrites 93.8%

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(\left(x \cdot c\right) \cdot x\right) \cdot s\right)}} \]

   if 1e167 < x

   1. Initial program 58.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 91.7

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 91.7%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      6. lift-/.f64 91.7

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      8. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      9. lift-+.f64 91.7

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      15. lower-*.f64 83.6

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      20. lift-*.f64 83.6

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

   6. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)} \]

      5. lower-*.f64 83.6

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)} \]

   8. Applied rewrites 83.6%

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right)} \cdot \left(x \cdot \left(\left(x \cdot c\right) \cdot s\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot c\right)\right) \cdot s\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot c\right)\right)\right) \cdot s}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(x \cdot c\right)\right)} \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot c\right)\right) \cdot s} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot c\right)\right) \cdot s} \]

      8. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(x \cdot \left(c \cdot s\right)\right)} \cdot \left(\left(x \cdot c\right) \cdot s\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(x \cdot \color{blue}{\left(c \cdot s\right)}\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(x \cdot c\right) \cdot s\right)} \cdot \left(\left(x \cdot c\right) \cdot s\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(x \cdot c\right)} \cdot s\right) \cdot \left(\left(x \cdot c\right) \cdot s\right)} \]

      13. swap-sqr N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \color{blue}{\left(s \cdot s\right)}} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)}} \]

      16. lower-*.f64 83.1

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right)} \cdot \left(s \cdot s\right)} \]

   10. Applied rewrites 83.1%

       \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 10^{+167}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(s \cdot \left(x \cdot \left(x \cdot c\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(x \cdot c\right) \cdot \left(x \cdot c\right)\right) \cdot \left(s \cdot s\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{+17}:\\ \;\;\;\;\frac{1}{s\_m \cdot \left(c\_m \cdot \left(c\_m \cdot \left(x\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right) \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0
         (/
          (cos (* x_m 2.0))
          (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))))
   (if (<= t_0 -2e-138)
     (/ -2.0 (* c_m (* c_m (* s_m s_m))))
     (if (<= t_0 1e+17)
       (/ 1.0 (* s_m (* c_m (* c_m (* x_m (* x_m s_m))))))
       (/ 1.0 (* x_m (* (* c_m (* x_m s_m)) (* c_m s_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))));
	double tmp;
	if (t_0 <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else if (t_0 <= 1e+17) {
		tmp = 1.0 / (s_m * (c_m * (c_m * (x_m * (x_m * s_m)))));
	} else {
		tmp = 1.0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))
    if (t_0 <= (-2d-138)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else if (t_0 <= 1d+17) then
        tmp = 1.0d0 / (s_m * (c_m * (c_m * (x_m * (x_m * s_m)))))
    else
        tmp = 1.0d0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))));
	double tmp;
	if (t_0 <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else if (t_0 <= 1e+17) {
		tmp = 1.0 / (s_m * (c_m * (c_m * (x_m * (x_m * s_m)))));
	} else {
		tmp = 1.0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))
	tmp = 0
	if t_0 <= -2e-138:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	elif t_0 <= 1e+17:
		tmp = 1.0 / (s_m * (c_m * (c_m * (x_m * (x_m * s_m)))))
	else:
		tmp = 1.0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0)))))
	tmp = 0.0
	if (t_0 <= -2e-138)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	elseif (t_0 <= 1e+17)
		tmp = Float64(1.0 / Float64(s_m * Float64(c_m * Float64(c_m * Float64(x_m * Float64(x_m * s_m))))));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(Float64(c_m * Float64(x_m * s_m)) * Float64(c_m * s_m))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))));
	tmp = 0.0;
	if (t_0 <= -2e-138)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	elseif (t_0 <= 1e+17)
		tmp = 1.0 / (s_m * (c_m * (c_m * (x_m * (x_m * s_m)))));
	else
		tmp = 1.0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-138], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+17], N[(1.0 / N[(s$95$m * N[(c$95$m * N[(c$95$m * N[(x$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot \left(s \cdot x\right)\right) \cdot s} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{c}^{2}} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      16. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      18. lower-*.f64 85.4

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

   4. Applied rewrites 85.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      7. lower-*.f64 49.0

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-2}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      7. lower-*.f64 49.2

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   10. Applied rewrites 49.2%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < 1e17

   1. Initial program 77.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 72.8

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right) \cdot x}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot x} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(s \cdot x\right)} \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot x} \]

      9. associate-*l* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(x \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)} \cdot x} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(s \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right) \cdot x} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(s \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right) \cdot x} \]

      12. associate-*r* N/A

          \[\leadsto \frac{1}{\left(s \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot s\right)}\right)\right) \cdot x} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(s \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot s\right)\right)\right) \cdot x} \]

      14. associate-*l* N/A

          \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(\left(x \cdot \left(c \cdot c\right)\right) \cdot s\right)}\right) \cdot x} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{\left(s \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot s\right)\right) \cdot x} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(s \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot s\right)\right) \cdot x} \]

      17. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{s \cdot \left(\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right) \cdot x\right)}} \]

      18. associate-*r* N/A

          \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{1}{s \cdot \left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

   7. Applied rewrites 88.0%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(c \cdot \left(x \cdot \left(s \cdot x\right)\right)\right)\right) \cdot s}} \]

   if 1e17 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 51.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 72.3

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(c \cdot s\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot s\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot s\right)\right)} \]

      8. lower-*.f64 80.0

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(c \cdot s\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot s\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(c \cdot s\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(c \cdot s\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(c \cdot s\right)\right)} \]

      13. lift-*.f64 79.4

          \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(c \cdot s\right)\right)} \]

   7. Applied rewrites 79.4%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot s\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{elif}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 10^{+17}:\\ \;\;\;\;\frac{1}{s \cdot \left(c \cdot \left(c \cdot \left(x \cdot \left(x \cdot s\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(c\_m \cdot \left(\left(x\_m \cdot s\_m\right) \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m \cdot \left(\left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right) \cdot \left(c\_m \cdot s\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0
         (/
          (cos (* x_m 2.0))
          (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))))
   (if (<= t_0 -2e-138)
     (/ -2.0 (* c_m (* c_m (* s_m s_m))))
     (if (<= t_0 INFINITY)
       (/ 1.0 (* c_m (* c_m (* (* x_m s_m) (* x_m s_m)))))
       (/ 1.0 (* x_m (* (* c_m (* x_m s_m)) (* c_m s_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))));
	double tmp;
	if (t_0 <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	} else {
		tmp = 1.0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)));
	}
	return tmp;
}

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))));
	double tmp;
	if (t_0 <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	} else {
		tmp = 1.0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))
	tmp = 0
	if t_0 <= -2e-138:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	elif t_0 <= math.inf:
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))))
	else:
		tmp = 1.0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0)))))
	tmp = 0.0
	if (t_0 <= -2e-138)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	elseif (t_0 <= Inf)
		tmp = Float64(1.0 / Float64(c_m * Float64(c_m * Float64(Float64(x_m * s_m) * Float64(x_m * s_m)))));
	else
		tmp = Float64(1.0 / Float64(x_m * Float64(Float64(c_m * Float64(x_m * s_m)) * Float64(c_m * s_m))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))));
	tmp = 0.0;
	if (t_0 <= -2e-138)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	elseif (t_0 <= Inf)
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	else
		tmp = 1.0 / (x_m * ((c_m * (x_m * s_m)) * (c_m * s_m)));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-138], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(1.0 / N[(c$95$m * N[(c$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x$95$m * N[(N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot \left(s \cdot x\right)\right) \cdot s} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{c}^{2}} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      16. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      18. lower-*.f64 85.4

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

   4. Applied rewrites 85.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      7. lower-*.f64 49.0

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-2}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      7. lower-*.f64 49.2

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   10. Applied rewrites 49.2%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 78.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 95.0

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 95.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}\right)\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      13. lower-*.f64 76.7

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   7. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)} \]

      4. unswap-sqr N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      7. lower-*.f64 84.1

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(x \cdot s\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      10. lower-*.f64 84.1

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(x \cdot s\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

      13. lower-*.f64 84.1

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

   9. Applied rewrites 84.1%

      \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\right)} \]

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 0.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 47.3

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)}\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(c \cdot s\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot s\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot s\right)\right)} \]

      8. lower-*.f64 65.6

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(c \cdot s\right)\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(c \cdot s\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(c \cdot s\right)\right)} \]

      11. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(c \cdot s\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(c \cdot s\right)\right)} \]

      13. lift-*.f64 63.7

          \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(c \cdot s\right)\right)} \]

   7. Applied rewrites 63.7%

      \[\leadsto \frac{1}{x \cdot \color{blue}{\left(\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot s\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{elif}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2 + \frac{1}{x\_m \cdot x\_m}}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<=
        (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
        -2e-138)
     (/ (+ -2.0 (/ 1.0 (* x_m x_m))) (* c_m (* c_m (* s_m s_m))))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -2e-138) {
		tmp = (-2.0 + (1.0 / (x_m * x_m))) / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if ((cos((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-2d-138)) then
        tmp = ((-2.0d0) + (1.0d0 / (x_m * x_m))) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if ((Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -2e-138) {
		tmp = (-2.0 + (1.0 / (x_m * x_m))) / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if (math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -2e-138:
		tmp = (-2.0 + (1.0 / (x_m * x_m))) / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = Float64(Float64(-2.0 + Float64(1.0 / Float64(x_m * x_m))) / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if ((cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = (-2.0 + (1.0 / (x_m * x_m))) / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-138], N[(N[(-2.0 + N[(1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot \left(s \cdot x\right)\right) \cdot s} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{c}^{2}} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      16. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      18. lower-*.f64 85.4

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

   4. Applied rewrites 85.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      7. lower-*.f64 49.0

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}} \]

      2. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}}} - 2 \cdot \frac{1}{{c}^{2} \cdot {s}^{2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}} - \color{blue}{\frac{2 \cdot 1}{{c}^{2} \cdot {s}^{2}}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}} - \frac{\color{blue}{2}}{{c}^{2} \cdot {s}^{2}} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{2}} - 2}{{c}^{2} \cdot {s}^{2}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{{x}^{2}} - 2}{{c}^{2} \cdot {s}^{2}}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(2\right)\right)}}{{c}^{2} \cdot {s}^{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{{x}^{2}} + \color{blue}{-2}}{{c}^{2} \cdot {s}^{2}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 + \frac{1}{{x}^{2}}}}{{c}^{2} \cdot {s}^{2}} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\color{blue}{-2 + \frac{1}{{x}^{2}}}}{{c}^{2} \cdot {s}^{2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{-2 + \color{blue}{\frac{1}{{x}^{2}}}}{{c}^{2} \cdot {s}^{2}} \]

      12. unpow2 N/A

          \[\leadsto \frac{-2 + \frac{1}{\color{blue}{x \cdot x}}}{{c}^{2} \cdot {s}^{2}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-2 + \frac{1}{\color{blue}{x \cdot x}}}{{c}^{2} \cdot {s}^{2}} \]

      14. unpow2 N/A

          \[\leadsto \frac{-2 + \frac{1}{x \cdot x}}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

      15. associate-*l* N/A

          \[\leadsto \frac{-2 + \frac{1}{x \cdot x}}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-2 + \frac{1}{x \cdot x}}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{-2 + \frac{1}{x \cdot x}}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}} \]

      18. unpow2 N/A

          \[\leadsto \frac{-2 + \frac{1}{x \cdot x}}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      19. lower-*.f64 49.2

          \[\leadsto \frac{-2 + \frac{1}{x \cdot x}}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   10. Applied rewrites 49.2%

       \[\leadsto \color{blue}{\frac{-2 + \frac{1}{x \cdot x}}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 72.5

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right) \cdot \left(x \cdot x\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      11. swap-sqr N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      14. lower-*.f64 82.9

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      17. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      19. lift-*.f64 82.6

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

      22. associate-*r* N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

      24. lift-*.f64 84.7

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

   7. Applied rewrites 84.7%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2 + \frac{1}{x \cdot x}}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<=
        (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
        -2e-138)
     (/ -2.0 (* c_m (* c_m (* s_m s_m))))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if ((cos((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-2d-138)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if ((Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if (math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -2e-138:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if ((cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-138], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot \left(s \cdot x\right)\right) \cdot s} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{c}^{2}} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      16. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      18. lower-*.f64 85.4

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

   4. Applied rewrites 85.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      7. lower-*.f64 49.0

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-2}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      7. lower-*.f64 49.2

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   10. Applied rewrites 49.2%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 72.5

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right) \cdot \left(x \cdot x\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      11. swap-sqr N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      14. lower-*.f64 82.9

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      17. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      19. lift-*.f64 82.6

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

      22. associate-*r* N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

      24. lift-*.f64 84.7

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

   7. Applied rewrites 84.7%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
      -2e-138)
   (/ -2.0 (* c_m (* c_m (* s_m s_m))))
   (/ 1.0 (* (* c_m s_m) (* x_m (* s_m (* x_m c_m)))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-2d-138)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -2e-138:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(Float64(c_m * s_m) * Float64(x_m * Float64(s_m * Float64(x_m * c_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-138], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot \left(s \cdot x\right)\right) \cdot s} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{c}^{2}} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      16. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      18. lower-*.f64 85.4

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

   4. Applied rewrites 85.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      7. lower-*.f64 49.0

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-2}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      7. lower-*.f64 49.2

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   10. Applied rewrites 49.2%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 95.8

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 95.8%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      6. lift-/.f64 95.8

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      8. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      9. lift-+.f64 95.8

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      15. lower-*.f64 94.2

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      20. lift-*.f64 93.2

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

   6. Applied rewrites 93.2%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)} \]

      5. lower-*.f64 93.9

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)} \]

   8. Applied rewrites 93.9%

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}\right)} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1}}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(x \cdot c\right) \cdot s\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.4%

       \[\leadsto \frac{\color{blue}{1}}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(x \cdot c\right) \cdot s\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(c\_m \cdot \left(\left(x\_m \cdot s\_m\right) \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
      -2e-138)
   (/ -2.0 (* c_m (* c_m (* s_m s_m))))
   (/ 1.0 (* c_m (* c_m (* (* x_m s_m) (* x_m s_m)))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-2d-138)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -2e-138:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(c_m * Float64(Float64(x_m * s_m) * Float64(x_m * s_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-138], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(c$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot \left(s \cdot x\right)\right) \cdot s} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{c}^{2}} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      16. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      18. lower-*.f64 85.4

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

   4. Applied rewrites 85.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      7. lower-*.f64 49.0

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-2}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      7. lower-*.f64 49.2

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   10. Applied rewrites 49.2%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 95.8

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 95.8%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}\right)\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      13. lower-*.f64 70.6

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   7. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)} \]

      4. unswap-sqr N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      7. lower-*.f64 76.6

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(x \cdot s\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      10. lower-*.f64 76.6

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(x \cdot s\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

      13. lower-*.f64 76.6

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

   9. Applied rewrites 76.6%

      \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x\_m \cdot 2\right)}{{c\_m}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s\_m}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot \left(s\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* x_m 2.0)) (* (pow c_m 2.0) (* x_m (* x_m (pow s_m 2.0)))))
      -2e-138)
   (/ -2.0 (* c_m (* c_m (* s_m s_m))))
   (/ 1.0 (* c_m (* c_m (* s_m (* s_m (* x_m x_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((x_m * 2.0)) / (pow(c_m, 2.0) * (x_m * (x_m * pow(s_m, 2.0))))) <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((x_m * 2.0d0)) / ((c_m ** 2.0d0) * (x_m * (x_m * (s_m ** 2.0d0))))) <= (-2d-138)) then
        tmp = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
    else
        tmp = 1.0d0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((x_m * 2.0)) / (Math.pow(c_m, 2.0) * (x_m * (x_m * Math.pow(s_m, 2.0))))) <= -2e-138) {
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	} else {
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((x_m * 2.0)) / (math.pow(c_m, 2.0) * (x_m * (x_m * math.pow(s_m, 2.0))))) <= -2e-138:
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)))
	else:
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(x_m * 2.0)) / Float64((c_m ^ 2.0) * Float64(x_m * Float64(x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))));
	else
		tmp = Float64(1.0 / Float64(c_m * Float64(c_m * Float64(s_m * Float64(s_m * Float64(x_m * x_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((x_m * 2.0)) / ((c_m ^ 2.0) * (x_m * (x_m * (s_m ^ 2.0))))) <= -2e-138)
		tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
	else
		tmp = 1.0 / (c_m * (c_m * (s_m * (s_m * (x_m * x_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-138], N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * N[(s$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-138

   1. Initial program 77.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot \left(s \cdot x\right)\right) \cdot s} \]

      15. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{c}^{2}} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      16. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      18. lower-*.f64 85.4

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

   4. Applied rewrites 85.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

      7. lower-*.f64 49.0

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   7. Applied rewrites 49.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-2}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

      7. lower-*.f64 49.2

         \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   10. Applied rewrites 49.2%

       \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]

   if -2.00000000000000013e-138 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 63.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 95.8

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 95.8%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}\right)\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      13. lower-*.f64 70.6

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   7. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(x \cdot 2\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq -2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;{s\_m}^{2} \leq 2 \cdot 10^{+274}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{c\_m \cdot \left(\left(x\_m \cdot c\_m\right) \cdot \left(s\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= (pow s_m 2.0) 2e+274)
     (/ (cos (+ x_m x_m)) (* c_m (* (* x_m c_m) (* s_m (* x_m s_m)))))
     (/ 1.0 (* t_0 t_0)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (pow(s_m, 2.0) <= 2e+274) {
		tmp = cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if ((s_m ** 2.0d0) <= 2d+274) then
        tmp = cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (Math.pow(s_m, 2.0) <= 2e+274) {
		tmp = Math.cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if math.pow(s_m, 2.0) <= 2e+274:
		tmp = math.cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if ((s_m ^ 2.0) <= 2e+274)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(c_m * Float64(Float64(x_m * c_m) * Float64(s_m * Float64(x_m * s_m)))));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if ((s_m ^ 2.0) <= 2e+274)
		tmp = cos((x_m + x_m)) / (c_m * ((x_m * c_m) * (s_m * (x_m * s_m))));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[s$95$m, 2.0], $MachinePrecision], 2e+274], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(c$95$m * N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s #s(literal 2 binary64)) < 1.99999999999999984e274

   1. Initial program 69.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 96.9

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 96.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      6. lift-/.f64 96.9

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      8. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      9. lift-+.f64 96.9

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      15. lower-*.f64 94.4

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      20. lift-*.f64 93.6

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

   6. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(s \cdot c\right)} \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{s \cdot \left(c \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \color{blue}{\left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot c\right) \cdot \left(s \cdot x\right)\right)}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(c \cdot \left(\color{blue}{\left(c \cdot x\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      10. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \color{blue}{\left(\left(c \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot x\right)\right)}} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(s \cdot x\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(s \cdot x\right)\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{s \cdot \color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

      17. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot x\right) \cdot s\right)}} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(\left(s \cdot x\right) \cdot s\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(\left(s \cdot x\right) \cdot s\right)} \]

      20. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot \left(c \cdot x\right)\right)} \cdot \left(\left(s \cdot x\right) \cdot s\right)} \]

      21. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(\left(c \cdot x\right) \cdot \left(\left(s \cdot x\right) \cdot s\right)\right)}} \]

   8. Applied rewrites 87.9%

      \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{c \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}} \]

   if 1.99999999999999984e274 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 54.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 72.0

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right) \cdot \left(x \cdot x\right)} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      11. swap-sqr N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      14. lower-*.f64 84.6

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      17. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      19. lift-*.f64 84.7

          \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)} \]

      22. associate-*r* N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

      23. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)} \]

      24. lift-*.f64 91.3

          \[\leadsto \frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}} \]

   7. Applied rewrites 91.3%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{s}^{2} \leq 2 \cdot 10^{+274}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{c \cdot \left(\left(x \cdot c\right) \cdot \left(s \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x\_m \cdot s\_m\right)\\ \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot \left(s\_m \cdot \left(x\_m \cdot c\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x_m s_m))))
   (if (<= x_m 2.2e-86)
     (/ (/ 1.0 t_0) t_0)
     (/ (cos (+ x_m x_m)) (* (* c_m s_m) (* x_m (* s_m (* x_m c_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.2e-86) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c_m * (x_m * s_m)
    if (x_m <= 2.2d-86) then
        tmp = (1.0d0 / t_0) / t_0
    else
        tmp = cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double t_0 = c_m * (x_m * s_m);
	double tmp;
	if (x_m <= 2.2e-86) {
		tmp = (1.0 / t_0) / t_0;
	} else {
		tmp = Math.cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	t_0 = c_m * (x_m * s_m)
	tmp = 0
	if x_m <= 2.2e-86:
		tmp = (1.0 / t_0) / t_0
	else:
		tmp = math.cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(c_m * Float64(x_m * s_m))
	tmp = 0.0
	if (x_m <= 2.2e-86)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(c_m * s_m) * Float64(x_m * Float64(s_m * Float64(x_m * c_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	t_0 = c_m * (x_m * s_m);
	tmp = 0.0;
	if (x_m <= 2.2e-86)
		tmp = (1.0 / t_0) / t_0;
	else
		tmp = cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (s_m * (x_m * c_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 2.2e-86], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(s$95$m * N[(x$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002e-86

   1. Initial program 64.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot {x}^{2}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right) \cdot x}} \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(\left({c}^{2} \cdot {s}^{2}\right) \cdot x\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left({c}^{2} \cdot {s}^{2}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left({s}^{2} \cdot {c}^{2}\right)}\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {c}^{2}\right)\right)} \]

      11. associate-*l* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(s \cdot {c}^{2}\right)\right)}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(c \cdot c\right)}\right)\right)\right)} \]

      14. associate-*r* N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot c\right)}\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(s \cdot c\right)\right)}\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      18. lower-*.f64 67.6

          \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

   5. Applied rewrites 67.6%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \left(c \cdot \color{blue}{\left(c \cdot s\right)}\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \left(x \cdot \color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{x \cdot \color{blue}{\left(x \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot \left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(c \cdot \left(c \cdot s\right)\right)\right)} \cdot \left(x \cdot x\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\left(s \cdot \color{blue}{\left(c \cdot \left(c \cdot s\right)\right)}\right) \cdot \left(x \cdot x\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot c\right) \cdot \left(c \cdot s\right)\right)} \cdot \left(x \cdot x\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot s\right)} \cdot \left(c \cdot s\right)\right) \cdot \left(x \cdot x\right)} \]

      13. swap-sqr N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]

   7. Applied rewrites 83.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}} \]

   if 2.2000000000000002e-86 < x

   1. Initial program 66.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 95.9

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 95.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      6. lift-/.f64 95.9

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      8. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      9. lift-+.f64 95.9

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      15. lower-*.f64 92.9

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      20. lift-*.f64 91.9

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

   6. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot x\right) \cdot s\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)} \]

      5. lower-*.f64 92.9

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot c\right)} \cdot s\right)\right)} \]

   8. Applied rewrites 92.9%

      \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(x \cdot c\right) \cdot s\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{c\_m \cdot \left(c\_m \cdot \left(\left(x\_m \cdot s\_m\right) \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(c\_m \cdot s\_m\right) \cdot \left(x\_m \cdot \left(c\_m \cdot \left(x\_m \cdot s\_m\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= x_m 4e-126)
   (/ 1.0 (* c_m (* c_m (* (* x_m s_m) (* x_m s_m)))))
   (/ (cos (+ x_m x_m)) (* (* c_m s_m) (* x_m (* c_m (* x_m s_m)))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 4e-126) {
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	} else {
		tmp = cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (c_m * (x_m * s_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x_m <= 4d-126) then
        tmp = 1.0d0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))))
    else
        tmp = cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (c_m * (x_m * s_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (x_m <= 4e-126) {
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	} else {
		tmp = Math.cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (c_m * (x_m * s_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if x_m <= 4e-126:
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))))
	else:
		tmp = math.cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (c_m * (x_m * s_m))))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (x_m <= 4e-126)
		tmp = Float64(1.0 / Float64(c_m * Float64(c_m * Float64(Float64(x_m * s_m) * Float64(x_m * s_m)))));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(c_m * s_m) * Float64(x_m * Float64(c_m * Float64(x_m * s_m)))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (x_m <= 4e-126)
		tmp = 1.0 / (c_m * (c_m * ((x_m * s_m) * (x_m * s_m))));
	else
		tmp = cos((x_m + x_m)) / ((c_m * s_m) * (x_m * (c_m * (x_m * s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[x$95$m, 4e-126], N[(1.0 / N[(c$95$m * N[(c$95$m * N[(N[(x$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * s$95$m), $MachinePrecision] * N[(x$95$m * N[(c$95$m * N[(x$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999998e-126

   1. Initial program 64.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 95.3

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 95.3%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left({s}^{2} \cdot {x}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{c \cdot \left(\left({s}^{2} \cdot {x}^{2}\right) \cdot c\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \color{blue}{\left(c \cdot \left({s}^{2} \cdot {x}^{2}\right)\right)}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot {x}^{2}\right)\right)} \]

      9. associate-*l* N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot \left(s \cdot {x}^{2}\right)\right)}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \color{blue}{\left(s \cdot {x}^{2}\right)}\right)\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      13. lower-*.f64 66.8

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

   7. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)} \]

      4. unswap-sqr N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(s \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      7. lower-*.f64 75.0

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(s \cdot x\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(x \cdot s\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      10. lower-*.f64 75.0

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\color{blue}{\left(x \cdot s\right)} \cdot \left(s \cdot x\right)\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

      13. lower-*.f64 75.0

          \[\leadsto \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \color{blue}{\left(x \cdot s\right)}\right)\right)} \]

   9. Applied rewrites 75.0%

      \[\leadsto \frac{1}{c \cdot \left(c \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)}\right)} \]

   if 3.9999999999999998e-126 < x

   1. Initial program 66.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left({s}^{2} \cdot x\right)} \cdot x\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left({s}^{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{{c}^{2}} \cdot {s}^{2}\right) \cdot \left(x \cdot x\right)} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot \color{blue}{{s}^{2}}\right) \cdot \left(x \cdot x\right)} \]

      10. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(c \cdot s\right)}^{2}} \cdot \left(x \cdot x\right)} \]

      11. pow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(c \cdot s\right)}^{2} \cdot \color{blue}{{x}^{2}}} \]

      12. pow-prod-down N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      15. lower-*.f64 96.3

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

   4. Applied rewrites 96.3%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{\color{blue}{\cos \left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}}^{2}} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      6. lift-/.f64 96.3

         \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(2 \cdot x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      8. count-2 N/A

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      9. lift-+.f64 96.3

         \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{{\left(\left(c \cdot s\right) \cdot x\right)}^{2}}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(\left(c \cdot s\right) \cdot x\right)} \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      15. lower-*.f64 93.6

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(\left(c \cdot s\right) \cdot x\right)\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(\color{blue}{\left(c \cdot s\right)} \cdot x\right)\right)} \]

      18. associate-*r* N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \color{blue}{\left(s \cdot x\right)}\right)\right)} \]

      20. lift-*.f64 92.8

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \color{blue}{\left(c \cdot \left(s \cdot x\right)\right)}\right)} \]

   6. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.8% accurate, 12.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{-2}{c\_m \cdot \left(c\_m \cdot \left(s\_m \cdot s\_m\right)\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m) :precision binary64 (/ -2.0 (* c_m (* c_m (* s_m s_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / (c_m * (c_m * (s_m * s_m)));
}

Fortran

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, c_m, s_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (-2.0d0) / (c_m * (c_m * (s_m * s_m)))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / (c_m * (c_m * (s_m * s_m)));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / (c_m * (c_m * (s_m * s_m)))

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(-2.0 / Float64(c_m * Float64(c_m * Float64(s_m * s_m))))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = -2.0 / (c_m * (c_m * (s_m * s_m)));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(-2.0 / N[(c$95$m * N[(c$95$m * N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.3%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(x \cdot \left(x \cdot {s}^{2}\right)\right)}} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot {s}^{2}\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(x \cdot {s}^{2}\right)}} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{{s}^{2}}\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   9. associate-*r* N/A

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot s\right) \cdot s\right)}} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(x \cdot s\right)\right) \cdot s}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left({c}^{2} \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left({c}^{2} \cdot x\right) \cdot \left(s \cdot x\right)\right)} \cdot s} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left({c}^{2} \cdot x\right)} \cdot \left(s \cdot x\right)\right) \cdot s} \]

   15. lift-pow.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{{c}^{2}} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   16. unpow2 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\color{blue}{\left(c \cdot c\right)} \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   18. lower-*.f64 80.8

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(s \cdot x\right)}\right) \cdot s} \]

4. Applied rewrites 80.8%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1 + -2 \cdot {x}^{2}}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{-2 \cdot {x}^{2} + 1}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   2. unpow2 N/A

      \[\leadsto \frac{-2 \cdot \color{blue}{\left(x \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-2 \cdot x\right) \cdot x} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \left(-2 \cdot x\right)} + 1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, -2 \cdot x, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

   7. lower-*.f64 54.6

      \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot -2}, 1\right)}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

7. Applied rewrites 54.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot -2, 1\right)}}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot x\right)\right) \cdot s} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{{c}^{2} \cdot {s}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{-2}{\color{blue}{\left(c \cdot c\right)} \cdot {s}^{2}} \]

   3. associate-*l* N/A

      \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-2}{\color{blue}{c \cdot \left(c \cdot {s}^{2}\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{-2}{c \cdot \color{blue}{\left(c \cdot {s}^{2}\right)}} \]

   6. unpow2 N/A

      \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

   7. lower-*.f64 28.2

      \[\leadsto \frac{-2}{c \cdot \left(c \cdot \color{blue}{\left(s \cdot s\right)}\right)} \]

10. Applied rewrites 28.2%

    \[\leadsto \color{blue}{\frac{-2}{c \cdot \left(c \cdot \left(s \cdot s\right)\right)}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))